Interaction Between Skew-Partitions and 2-joins

One can think of algorithms for testing for odd holes if you use only
2-joins to decompose a graph, or if you use only skew partitions. However,
the interaction between skew partition steps and 2-join steps adds another
level of difficulty. Can one argue that this can be reduced only to
testing for holes as you decompose using skew partitions only and testing for
holes using 2-joins only?

Contributed by Jeremy Spinrad

Similar approach worked in algorithms for recognizing even-hole-free graphs:
first the graph is decomposed via vertex cut-sets, and only then via 2-joins.
The 2-join decomposition blocks are defined in such a way that no new vertex cut- set is introduced.